Sieve (category Theory) - Pullback of Sieves

Pullback of Sieves

The most common operation on a sieve is pullback. Pulling back a sieve S on c by an arrow f:c′→c gives a new sieve f*S on c′. This new sieve consists of all the arrows in S which factor through c′.

There are several equivalent ways of defining f*S. The simplest is:

For any object d of C, f*S(d) = { g:d→c′ | fg ∈ S(d)}

A more abstract formulation is:

f*S is the image of the fibered product S×_{Hom(−, c)}Hom(−, c′) under the natural projection S×_{Hom(−, c)}Hom(−, c′)→Hom(−, c′).

Here the map Hom(−, c′)→Hom(−, c) is Hom(f, c′), the pullback by f.

The latter formulation suggests that we can also take the image of S×_{Hom(−, c)}Hom(−, c′) under the natural map to Hom(−, c). This will be the image of f*S under composition with f. For each object d of C, this sieve will consist of all arrows fg, where g:d→c′ is an arrow of f*S(d). In other words, it consists of all arrows in S that can be factored through f.

If we denote by ∅_c the empty sieve on c, that is, the sieve for which ∅(d) is always the empty set, then for any f:c′→c, f*∅_c is ∅_c′. Furthermore, f*Hom(−, c) = Hom(−, c′).